Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-28T02:07:47.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method

Published online by Cambridge University Press:  10 January 2014

Fabien Casenave ,
Alexandre Ern  and
Tony Lelièvre
Fabien Casenave
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France
Alexandre Ern
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France
Tony Lelièvre
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France INRIA Rocquencourt, MICMAC Team-Project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
Get access

Abstract

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

Keywords

Reduced basis methoda posteriori error boundround-off errorsboundary element methodempirical interpolation methodacoustics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 207 - 229
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bai, Z. and Skoogh, D., Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9–44. Google Scholar
Bahayou, M.A., Sur le problème de Helmholtz. Rendiconti del Seminario matematico della Università e Politecnico di Torino (2007) 427–450. Google Scholar
Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. Google Scholar
Björck, A. and Paige, C.C., Loss and recapture of orthogonality in the modified Gram–Schmidt algorithm. SIAM J. Matrix Anal. Appl. 13 (1992) 176–190. Google Scholar
S. Boyaval, Mathematical modelling and numerical simulation in materials science. Ph.D. thesis, Université Paris-Est (2009).
Buffa, A. and Hiptmair, R., Regularized combined field integral equations. Numer. Math. 100 (2005) 1–19. Google Scholar
R.L. Burden and J.D. Faires, Numerical Analysis. PWS Publishing Company (1993).
Cancès, E., Ehrlacher, V. and Lelièvre, T., Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433–2467. Google Scholar
Casenave, F., Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542. Google Scholar
F. Casenave, Ph.D. thesis, in preparation (2013).
Casenave, F., Ghattassi, M. and Joubaud, R., A multiscale problem in thermal science. ESAIM: Proceedings 38 (2012) 202–219. Google Scholar
Chatterjee, A., An introduction to the proper orthogonal decomposition. Curr. Sci. 78 (2000) 808–817. Google Scholar
Y. Chen, J.S. Hesthaven, Y. Maday, J. Rodriguez and X. Zhu, Certified reduced basis method for electromagnetic scattering and radar cross section estimation. Technical Report 2011-28, Scientific Computing Group, Brown University, Providence, RI, USA (2011).
Chen, Y., Hesthaven, J.S., Maday, Y. and Rodríguez, J., Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem. ESAIM: M2AN 43 (2009) 1099–1116. Google Scholar
Chinesta, F., Ladeveze, P. and Elías, C., A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Eng. 18 (2011) 395–404. Google Scholar
A. Delnevo, I. Terrasse, Code ACTI3S harmonique : Justifications Mathématiques : Partie I. Technical report, EADS CCR (2001).
A. Delnevo, I. Terrasse, Code ACTI3S, Justifications Mathématiques : Partie II, présence d’un écoulement uniforme. Technical report, EADS CCR (2002).
A. Ern and J.L. Guermond, Theory and Practice of Finite Elements, in vol. 159 of Applied Mathematical Sciences. Springer (2004).
Fares, M., Hesthaven, J.S., Maday, Y. and Stamm, B., The reduced basis method for the electric field integral equation. J. Comput. Phys. 230 (2011) 5532–5555. Google Scholar
Giraud, L. and Langou, J., When modified Gram–Schmidt generates a well-conditioned set of vectors. IMA J. Numer. Anal. 22 (2002) 521–528. Google Scholar
Goldberg, D., What every computer scientist should know about floating point arithmetic. ACM Computing Surveys 23 (1991) 5–48. Google Scholar
G.H. Golub and C.F. Van Loan, Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press (1996).
Guyan, R.J., Reduction of stiffness and mass matrices. AIAA J. 3 (1965) 380. Google Scholar
Hiptmair, R., Coercive combined field integral equations. J. Numer. Math. 11 (2003) 115–134. Google Scholar
R. Hiptmair and P. Meury, Stable FEM-BEM Coupling for Helmholtz Transmission Problems. ETH, Seminar für Angewandte Mathematik (2005).
G.C. Hsiao and W.L. Wendland, Boundary Element Methods: Foundation and Error Analysis. John Wiley & Sons, Ltd (2004).
Huynh, D.B.P., Rozza, G., Sen, S. and Patera, A.T., A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. Acad. Sci. Paris 345 (2007) 473–478. Google Scholar
P. Langlois, S. Graillat and N. Louvet, Compensated Horner scheme. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006).
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D.V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Math. Acad. Sci. Paris 331 (2000) 153–158. Google Scholar
Maday, Y., Nguyen, N.C., Patera, A.T. and Pau, S., A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2008) 383–404. Google Scholar
W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000).
Nouy, A. and Le Maître, O.P., Generalized spectral decomposition for stochastic nonlinear problems. J. Comput. Phys. 228 (2009) 202–235. Google Scholar
A.T. Patera, Private communication (2012).
A.T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2007).
Paz, M., Dynamic condensation. AIAA J. 22 (1984) 724–727. Google Scholar
Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124 (2002) 70–80. Google Scholar
S.A. Sauter and C. Schwab, Boundary Element Methods. Springer Series in Computational Mathematics. Springer (2010).
I.E. Shparlinski, Sparse polynomial approximation in finite fields. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, STOC ’01. ACM, New York, USA (2001) 209–215.
Veroy, K. and Patera, A.T., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47 (2005) 773–788. Google Scholar
Veroy, K., Prud’homme, C. and Patera, A.T., Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Math. Acad. Sci. Paris 337 (2003) 619–624. Google Scholar